5
Steel elements
5.1 Structural design
At present there are two British Standards devoted to the design ofstruc-of steelwork
tural steel elements:BS 449 The use of structural steel in building. BS 5950 Structural use of steelwork in building.
The former employs permissible stress analysis whilst the latter is based upon limit state philosophy. Since it is intended that BS 5950 will eventu-ally replace BS 449, the designs contained in this manual will be based upon BS 5950.
There are to be nine parts to BS 5950: Part 1 Part 2 Part 3 Part 4 Part 5 Part 6 Part 7 Part 8 Part 9
Code of practice for design in simple and continuous construc-tion: hot rolled sections.
Specification for materials, fabrication and erection: hot rolled sections.
Code of practice for design in composite construction.
Code of practice for design of floors with profiled steel sheeting. Code of practice for design of cold formed sections.
Code of practice for design in light gauge sheeting, decking and cladding.
Specification for materials and workmanship: cold formed sections.
Code of practice for design of fire protection for structural steelwork.
Code of practice for stressed skin design.
Calculations for the majority of steel members contained in building and allied structures are usually based upon the guidance given in Part 1 of the standard. This manual will therefore be related to that part.
Requirements for the fabrication and erection of structural steelwork are given in Part 2 of the standard. The designer should also be familiar with these, so that he can take into account any which could influence his design.
For information on all aspects of bridge design, reference should be made to BS 5400, ‘Steel, concrete and composite bridges’.
5.2 Symbols
The design of a steel structure may be divided into two stages. First the size of the individual members is determined in relation to the induced forces and bending moments. Then all necessary bolted or welded connec-tions are designed so that they are capable of transmitting the forces and bending moments. In this manual we will concentrate on the design of the main structural elements.
Three methods of design are included in BS 5950 Part 1:
Simple design This method applies to structures in which the end
connec-tions between members are such that they cannot develop any significant restraint moments. Thus, for the purpose of design, the structure may be considered to be pin-jointed on the basis of the following assumptions: (a) All beams are simply supported.
(b) All connections are designed to resist only resultant reactions at the appropriate eccentricity.
(c) Columns are subjected to loads applied at the appropriate eccentricity. (d) Resistance to sway, such as that resulting from lateral wind loads, is
provided by either bracing, shear walls or core walls.
Rigid design In this method the structure is considered to be rigidly
jointed such that it behaves as a continuous framework. Therefore the connections must be capable of transmitting both forces and bending moments. Portal frames are designed in this manner using either elastic or plastic analysis.
Semi-rigid design This is an empirical method, seldom adopted, which
permits partial interaction between beams and columns to be assumed provided that certain stated parameters are satisfied.
The design of steel elements dealt with in this manual will be based upon the principles of simple design.
It is important to appreciate that an economic steel design is not neces-sarily that which uses the least weight of steel. The most economical solution will be that which produces the lowest overall cost in terms of materials, detailing, fabrication and erection.
The symbols used in BS 5950 and which are relevant to this manual are as follows:
A area
Ag gross sectional area of steel section
Av shear area (sections) B breadth of section b outstand of flange b1 stiff bearing length
d E Fc Fv Ix Iy L LE M MA Mb Mcx, Mcy Me Mo Mu Mx M m Pc Pcrip Pv pb Pw py rx, ry Sx, Sy T t u v x Zx, Zy ß f e m δ ε depth of web
modulus of elasticity of steel eccentricity
ultimate applied axial load shear force (sections)
second moment of area about the major axis second moment of area about the minor axis length of span
effective length larger end moment
maximum moment on the member or portion of the member under consideration
buckling resistance moment (lateral torsional)
moment capacity of section about the major and minor axes in the absence of axial load
eccentricity moment
mid-length moment on a simply supported span equal to the unrestrained length
ultimate moment
maximum moment occurring between lateral restraints on a beam
equivalent uniform moment equivalent uniform moment factor slenderness correction factor compression resistance of column ultimate web bearing capacity shear capacity of a section bending strength
compressive strength
buckling resistance of an unstiffened web design strength of steel
radius of gyration of a member about its major and minor axes
plastic modulus about the major and minor axes thickness of a flange or leg
thickness of a web or as otherwise defined in a clause buckling parameter of the section
slenderness factor for beam torsional index of section
elastic modulus about the major and minor axes ratio of smaller to larger end moment
overall load factor
load variation factor: function of e1 and e2
material strength factor
ratio M/M0, that is the ratio of the larger end moment to the
mid-length moment on a simply supported span equal to the unrestrained length deflection constant (275/ py) 1 / 2 e n pc γ γ γ γ γ γ
5.3 Definitions
slenderness, that is the effective length divided by the radius of gyration
LT equivalent slenderness
The following definitions which are relevant to this manual have been abstracted from BS 5950 Part 1:
Beam A member predominantly subject to bending.
Buckling resistance Limit of force or moment which a member can
with-stand without buckling.
Capacity Limit of force or moment which may be applied without
caus-ing failure due to yieldcaus-ing or rupture.
Column A vertical member of a structure carrying axial load and
poss-ibly moments.
Compact cross-section A cross-section which can develop the plastic
moment capacity of the section but in which local buckling prevents rotation at constant moment.
Dead load All loads of constant magnitude and position that act
perman-ently, including self-weight.
Design strength The yield strength of the material multiplied by the
ap-propriate partial factor.
Effective length Length between points of effective restraint of a member
multiplied by a factor to take account of the end conditions and loading.
Elastic design Design which assumes no redistribution of moments due
to plastic rotation of a section throughout the structure.
Empirical method Simplified method of design justified by experience or
testing.
Factored load Specified load multiplied by the relevant partial factor. H-section A section with one central web and two equal flanges which
has an overall depth not greater than 1.2 times the width of the flange.
I-section Section with central web and two equal flanges which has an
overall depth greater than 1.2 times the width of the flange.
Imposed load Load on a structure or member other than wind load,
produced by the external environment and intended occupancy or use.
Lateral restraint For a beam: restraint which prevents lateral movement
of the compression flange. For a column: restraint which prevents lateral movement of the member in a particular plane.
Plastic cross-section A cross-section which can develop a plastic hinge
with sufficient rotation capacity to allow redistribution of bending moments within the structure.
Plastic design Design method assuming redistribution of moment in
continuous construction.
λ λ
Semi-compact cross-section A cross-section in which the stress in the
ex-treme fibres should be limited to yield because local buckling would pre-vent development of the plastic moment capacity in the section.
Serviceability limit states Those limit states which when exceeded can
lead to the structure being unfit for its intended use.
Slender cross-section A cross-section in which yield of the extreme fibres
cannot be attained because of premature local buckling.
Slenderness The effective length divided by the radius of gyration. Strength Resistance to failure by yielding or buckling.
Strut A member of a structure carrying predominantly compressive axial
load.
Ultimate limit state That state which if exceeded can cause collapse of
part or the whole of the structure.
5.4 Steel grades and
sections
As mentioned in Chapter 1, steel sections are produced by rolling the steel, whilst hot, into various standard profiles. The quality of the steel that is used must comply with BS 4360 ‘Specification for weldable structural steels’, which designates four basic grades for steel: 40, 43, 50 and 55. (It should be noted that grade 40 steel is not used for structural purposes.) These basic grades are further classified in relation to their ductility, de-noted by suffix letters A, B, C and so on. These in turn give grades 43A, 43B, 43C and so on. The examples in this manual will, for simplicity, be based on the use of grade 43A steel.
It is eventually intended to replace the present designations with grade references related to the yield strength of the steel. Thus, for example, grade 43A steel will become grade 275A since it has a yield stress of 275 N/mm2.
The dimensions and geometric properties of the various hot rolled sections are obtained from the relevant British Standards. Those for uni-versal beam (UB) sections, uniuni-versal column (UC) sections, rolled steel joist (RSJ) sections and rolled steel channel (RSC) sections are given in BS 4 Part 1. Structural hollow sections and angles are covered by BS 4848 Part 2 and Part 4 respectively. It is eventually intended that BS 4 Part 1 will also become part of BS 4848.
Cold formed steel sections produced from light gauge plate, sheet or strip are also available. Their use is generally confined to special applica-tions and the production of proprietary roof purlins and sheeting rails. Guidance on design using cold formed sections is given in BS 5950 Part 5.
5.5 Design philosophy
The design approach employed in BS 5950 is based on limit state philo-sophy. The fundamental principles of the philosophy were explained in Chapter 3 in the context of concrete design. In relation to steel structures, some of the ultimate and serviceability limit states (ULSs and SLSs) that may have to be considered are as followsUltimate limit states
Strength The individual structural elements should be checked to ensure
that they will not yield, rupture or buckle under the influence of the ultimate design loads, forces, moments and so on. This will entail checking beams for the ULSs of bending and shear, and columns for a compressive ULS and when applicable a bending ULS.
Stability The building or structural framework as a whole should be
checked to ensure that the applied loads do not induce excessive sway or cause overturning.
Fracture due to fatigue Fatigue failure could occur in a structure that is
repeatedly subjected to rapid reversal of stress. Connections are particu-larly prone to such failure. In the majority of building structures, changes in stress are gradual. However, where dynamic loading could occur, such as from travelling cranes, the risk of fatigue failure should be considered.
Brittle failure Sudden failure due to brittle fracture can occur in
steel-work exposed to low temperatures; welded structures are particularly susceptible. Since the steel members in most building frames are protected from the weather, they are not exposed to low temperatures and therefore brittle fracture need not be considered. It is more likely to occur in large welded structures, such as bridges, which are exposed to the extremes of winter temperature. In such circumstances, it is necessary to select steel of adequate notch ductility and to devise details that avoid high stress concentrations.
Serviceability limit states
Deflection Adequate provision must be made to ensure that excessive
deflection which could adversely effect any components or finishes sup-ported by the steel members does not occur.
Corrosion and durability Corrosion induced by atmospheric or chemical
conditions can adversely affect the durability of a steel structure. The designer must therefore specify a protective treatment suited to the loca-tion of the structure. Guidance on the selecloca-tion of treatments is given in BS 5493 ‘Code of practice for protective coating of iron and steel structures against corrosion’. Certain classes of grade 50 steel are also available with weather resistant qualities, indicated by the prefix WR, for example WR 50A. Such steel when used in a normal external environment does not need any additional surface protection. An oxide skin forms on the surface of the steel, preventing further corrosion. Provided that the self-coloured appearance is aesthetically acceptable, consideration may be given to its use in situations where exposed steel is permitted, although it should be borne in mind that it is more expensive than ordinary steel.
Fire protection Due consideration should also be given to the provision
of adequate protection to satisfy fire regulations. Traditionally fire protec-tion was provided by casing the steelwork in concrete. Nowadays a num-ber of lightweight alternatives are available in the form of dry sheet
material, plaster applied to metal lathing, or plaster sprayed directly on to the surface of the steel. Intumescent paints are also marketed which froth when heated to produce a protective insulating layer on the surface of the steel.
Since this manual is concerned with the design of individual structural elements, only the strength ULS and the deflection SLS will be considered further.
5.6 Safety factors
In a similar fashion to concrete and masonry design, partial safety factors are once again applied separately to the loads and material stresses. Ini-tially BS 5950 introduces a third factor, p, related to structuralperform-ance. The factors given in BS 5950 are as follows:
e for load
p for structural performance m for material strength.
However, factors e and p when multiplied together give a single partial
safety factor for load of f. Hence the three partial safety factors reduce to the usual two of f and m.
5.7 Loads
The basic loads are referred to in BS 5950 as specified loads rather than characteristic loads. They need to be multiplied by the relevant partial safety factor for load f to arrive at the design load.5.7.1 Specified loads
These are the same as the characteristic loads of dead, imposed and wind previously defined in Chapters 3 and 4 in the context of concrete and masonry design.
5.7.2 Partial safety factors for load
To arrive at the design load, the respective specified loads are multiplied by a partial safety factor f in relation to the limit state being considered:
Design load = f × specified load
5.7.3 Ultimate design load
The partial safety factors for the ULS load combinations are given in Table 2 of BS 5950. For the beam and column examples contained in this
γ γ γ γ γ γ γ γ γ γ γ γ
manual, only the values for the dead and imposed load combination are required, which are 1.4 and 1.6 respectively. Thus the ultimate design load for the dead plus imposed combination would be as follows:
Ultimate design load = f× dead load + f× imposed load
= 1.4 × dead load + 1.6 × imposed load
5.7.4 Serviceability design load
For the purpose of checking the deflection SLS, the partial safety factor
f may be taken as unity. Furthermore, in accordance with BS 5950, the
deflection of a beam need only be checked for the effect of imposed load-ing. Hence the serviceability design load for checking the deflection of a steel beam is simply the specified imposed load. This differs from the design of timber and concrete beams, for which the dead plus imposed load is used to check deflection. However, it is not unreasonable since we are only interested in controlling the deflection of steel beams to avoid damage to finishes, and the dead load deflection will already have taken place before these are applied. If for reasons of appearance it is considered necessary to counteract all or part of the dead load deflection, the beam could be pre-cambered.
5.8 Material properties
The ultimate design strength py for the most common types of structuralsteel are given in BS 5950 Table 6, from which those for grade 43 steel are shown here in Table 5.1. They incorporate the material partial safety factor m in the specified values. Therefore the strength may be obtained
directly from the table without further modification. For beam and col-umn sections the material thickness referred to in the table should be taken as the flange thickness.
Table 5.1 Design strength py of grade 43 steel
Thickness less than or equal to (mm)
py for rolled sections, plates and hollow
sections (N/mm2) 16 275 40 265 63 255 100 245
The modulus of elasticity E, for deflection purposes, may be taken as 205 kN/mm2 for all grades of steel.
γ γ
γ
5.9 Section properties
Dimensions and geometric properties for the hot rolled steel sections commonly available for use as beams and columns are tabulated in BS 4 Part 1. Similar tables expanded to include a number of useful design constants are also published by the Steel Construction Institute. These are contained in their Steelwork Design Guide to BS 5950: Part 1, Vol-ume 1, Section Properties, Member Capacities. Tables 5.2 and 5.3 given here are typical examples from that publication, reproduced by kind per-mission of the director of the Steel Construction Institute. Complete copies of the guide can be obtained from the Institute at Silwood Park, Ascot, Berkshire, SL5 7QN.Table 5.2 relates to universal beam (UB) sections, as illustrated in Figure 5.1, and Table 5.3 to universal column (UC) sections, as illustrated in Figure 5.2. The use of these tables in relation to the design of beams and columns will be explained in the appropriate sections of this chapter. Whilst the UB sections are primarily intended for use as beams, they can if desired be used as columns; this is often the case in portal frame construction. Similarly the UC sections are intended for use as columns but can also be used as beams. However, because they have a stocky cross-section they do not lend themselves as readily to such an alternative use. B y y B Flange Flange T T Web Web D x x d D x x d t t Flange b b y y
Figure 5.1 Universal beam cross-section Figure 5.2 Universal column cross-section
5.10 Beams
The main structural design requirements for which steel beams should be examined as as follows:(a) Bending ULS
Table 5.2 Universal beams (abstracted from the Steelwork Design Guide to BS 59.50: Part 1, published by the Steel
Construction Institute) (a) Dimensions
Designation Depth Width Thickness Root Depth Ratios for Dimensions for detailing Surface area of of radius between local buckling
Serial Mass section section Web Flange fillets Flange Web End Notch Per per
size per D B r d b/T d/t clearance metre tonne
metre t T C N n (mm) (kg) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (m2) (m2) 914 × 419 388 920.5 420.5 21.5 36.6 24.1 799.1 5.74 37.2 13 210 62 3.44 8.86 343 911.4 418.5 19.4 32.0 24.1 799.1 6.54 41.2 12 210 58 3.42 9.96 914 × 305 289 926.6 307.8 19.6 32.0 19.1 824.5 4.81 42.1 12 156 52 3.01 10.4 253 918.5 305.5 17.3 27.9 19.1 824.5 5.47 47.7 11 156 48 2.99 11.8 224 910.3 304.1 15.9 23.9 19.1 824.5 6.36 51.9 10 156 44 2.97 13.3 201 903.0 303.4 15.2 20.2 19.1 824.5 7.51 54.2 10 156 40 2.96 14.7 838 × 292 226 850.9 293.8 16.1 26.8 17.8 761.7 5.48 47.3 10 150 46 2.81 12.5 194 840.7 292.4 14.7 21.7 17.8 761.7 6.74 51.8 9 150 40 2.79 14.4 176 834.9 291.6 14.0 18.8 17.8 761.7 7.76 54.5 9 150 38 2.78 15.8 762 × 267 197 769.6 268.0 15.6 25.4 16.5 685.8 5.28 44.0 10 138 42 2.55 13.0 173 762.0 266.7 14.3 21.6 16.5 685.8 6.17 48.0 9 138 40 2.53 14.6 147 753.9 265.3 12.9 17.5 16.5 685.8 7.58 53.2 8 138 36 2.51 17.1 686 × 254 170 692.9 255.8 14.5 23.7 15.2 615.1 5.40 42.4 9 132 40 2.35 13.8 152 687.6 254.5 13.2 21.0 15.2 615.1 6.06 46.6 9 132 38 2.34 15.4 140 683.5 253.7 12.4 19.0 15.2 615.1 6.68 49.6 8 132 36 2.33 16.6 125 677.9 253.0 11.7 16.2 15.2 615.1 7.81 52.6 8 132 32 2.32 18.5 610 × 305 238 633.0 311.5 18.6 31.4 16.5 537.2 4.96 28.9 11 158 48 2.45 10.3 179 617.5 307.0 14.1 23.6 16.5 537.2 6.50 38.1 9 158 42 2.41 13.4 149 609.6 304.8 11.9 19.7 16.5 537.2 7.74 45.1 8 158 38 2.39 16.0 610 × 229 140 617.0 230.1 13.1 22.1 12.7 547.3 5.21 41.8 9 120 36 2.11 15.0 125 611.9 229.0 11.9 19.6 12.7 547.3 5.84 46.0 8 120 34 2.09 16.8 113 607.3 228.2 11.2 17.3 12.7 547.3 6.60 48.9 8 120 32 2.08 18.4 101 602.2 227.6 10.6 14.8 12.7 547.3 7.69 51.6 7 120 28 2.07 20.5 533 × 210 122 544.6 211.9 12.8 21.3 12.7 476.5 4.97 37.2 8 110 36 1.89 15.5 109 539.5 210.7 11.6 18.8 12.7 476.5 5.60 41.1 8 110 32 1.88 17.2 101 536.7 210.1 10.9 17.4 12.7 476.5 6.04 43.7 7 110 32 1.87 18.5 92 533.1 209.3 10.2 15.6 12.7 476.5 6.71 46.7 7 110 30 1.86 20.2 82 528.3 208.7 9.6 13.2 12.7 476.5 7.91 49.6 7 110 26 1.85 22.6 457 × 191 98 467.4 192.8 11.4 19.6 89 463.6 192.0 10.6 17.7 82 460.2 191.3 9.9 16.0 74 457.2 190.5 9.1 14.5 67 453.6 189.9 8.5 12.7 4.92 35.8 8 102 30 1.67 17.0 5.42 38.5 7 102 28 166 18.6 5.98 41.2 7 102 28 1.65 20.1 6.57 44.8 7 102 26 1.64 22.2 7.48 48.0 6 102 24 1.63 24.4 457 × 152 82 465.1 153.5 10.7 18.9 74 461.3 152.7 9.9 17.0 67 457.3 151.9 9.1 15.0 60 454.7 152.9 8.0 13.3 52 449.8 152.4 7.6 10.9 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 407.9 407.9 407.9 407.9 407.9 407.0 407.0 407.0 407.0 407.0 4.06 38.0 7 82 30 1.51 18.4 4.49 41.1 7 82 28 1.50 20.2 5.06 44.7 7 82 26 1.49 22.2 5.75 51.0 6 84 24 1.49 24.8 6.99 53.6 6 84 22 1.48 28.4
Table 5.2 Universal beams continued (abstracted from the Steelwork Design Guide to BS 5950: Part 1, published by
the Steel Construction Institute) (b) Properties
Designation Second moment Radius Elastic Plastic Buckling Torsional Warping Torsional Area of area of gyration modulus modulus parameter index constant constant of
Serial Mass Axis Axis Axis Axis Axis Axis Axis Axis section
size per x–x y–y x–x y–y x–x y–y x–x y–y u x H J A
metre (mm) (kg) (cm4) (cm4) (cm) (cm) (cm3) (cm3) (cm3) (cm3) (dm6) (cm4) (cm2) 914 × 419 388 343 914 × 305 289 253 224 201 838 × 292 226 194 176 762 × 267 197 173 147 686 × 254 170 152 140 125 610 × 305 238 179 149 610 × 229 140 125 113 101 533 × 210 122 109 101 92 82 457 × 191 98 89 82 74 67 457 × 152 82 74 67 60 52 719 000 45 400 625 000 39 200 505 000 15 600 437 000 13 300 376 000 11 200 326 000 9 430 340 000 11 400 279 000 9 070 246 000 7 790 240 000 8 170 205 000 6 850 169 000 5 470 170 000 6 620 150 000 5 780 136 000 5 180 118 000 4 380 208 000 15 800 152 000 11 400 125 000 9 300 112 000 4 510 98 600 3 930 87 400 3 440 75 700 2 910 76 200 3 390 66 700 2 940 61 700 2 690 55 400 2 390 47 500 2 010 45 700 2 340 41 000 2 090 37 100 1 870 33 400 1 670 29 400 1 450 36 200 1 140 32 400 1 010 28 600 878 25 500 794 21 300 645 38.1 9.58 15 600 2160 17 700 3340 0.884 26.7 88.7 1730 494 37.8 9.46 13 700 1870 15 500 2890 0.883 30.1 75.7 1190 437 37.0 6.51 10 900 1010 12 600 1600 0.867 31.9 31.2 929 369 36.8 6.42 9 510 872 10 900 1370 0.866 36.2 26.4 627 323 36.3 6.27 8 260 738 9 520 1160 0.861 41.3 22.0 421 285 35.6 6.06 7 210 621 8 360 983 0.853 46.8 18.4 293 256 34.3 6.27 7 990 773 9 160 1210 0.87 35.0 19.3 514 289 33.6 6.06 6 650 620 7 650 974 0.862 41.6 15.2 307 247 33.1 5.90 5 890 534 6 810 842 0.856 46.5 13.0 222 224 30.9 5.71 6 230 610 7 170 959 0.869 33.2 11.3 405 251 30.5 5.57 5 390 513 6 200 807 0.864 38.1 9.38 267 220 30.0 5.39 4 480 412 5 170 649 0.857 45.1 7.41 161 188 28.0 5.53 4 910 518 5 620 810 0.872 31.8 7.41 307 217 27.8 5.46 4 370 454 5 000 710 0.871 35.5 6.42 219 194 27.6 5.38 3 990 408 4 560 638 0.868 38.7 5.72 169 179 27.2 5.24 3 480 346 4 000 542 0.862 43.9 4.79 116 160 26.1 7.22 6 560 1020 7 460 1570 0.886 21.1 14.3 788 304 25.8 7.08 4 910 743 5 520 1140 0.886 27.5 10.1 341 228 25.6 6.99 4 090 610 4 570 937 0.886 32.5 8.09 200 190 25.0 5.03 3 630 392 4 150 612 0.875 30.5 3.99 217 178 24.9 4.96 3 220 344 3 680 536 0.873 34.0 3.45 155 160 24.6 4.88 2 880 301 3 290 470 0.87 37.9 2.99 112 144 24.2 4.75 2 510 256 2 880 400 0.863 43.0 2.51 77.2 129 22.1 4.67 2 800 320 3 200 501 0.876 27.6 2.32 180 156 21.9 4.60 2 470 279 2 820 435 0.875 30.9 1.99 126 139 21.8 4.56 2 300 257 2 620 400 0.874 33.1 1.82 102 129 21.7 4.51 2 080 229 2 370 356 0.872 36.4 1.60 76.2 118 21.3 4.38 1 800 192 2 060 300 0.865 41.6 1.33 51.3 104 19.1 4.33 1 960 243 2 230 378 0.88 25.8 1.17 121 125 19.0 4.28 1 770 217 2 010 338 0.879 28.3 1.04 90.5 114 18.8 4.23 1 610 196 1 830 304 0.877 30.9 0.923 69.2 105 18.7 4.19 1 460 175 1 660 272 0.876 33.9 0.819 52.0 95.0 18.5 4.12 1 300 153 1 470 237 0.873 37.9 0.706 37.1 85.4 18.6 3.31 1 560 149 1 800 235 0.872 27.3 0.569 89.3 104 18.5 3.26 1 410 133 1 620 209 0.87 30.0 0.499 66.6 95.0 18.3 3.21 1 250 116 1 440 182 0.867 33.6 0.429 47.5 85.4 18.3 3.23 1 120 104 1 280 163 0.869 37.5 0.387 33.6 75.9 17.9 3.11 949 84.6 1 090 133 0.859 43.9 0.311 21.3 66.5
Table 5.3 Universal columns (abstracted from the Steelwork Design Guide to BS 5950: Part 1, published by the Steel
Construction Institute) (a) Dimensions
Designation Depth Width Thickness Root Depth Ratios for Dimensions for detailing Surface area Of Of radius between local buckling
Serial Mass section section Web Flange fillets Flange Web End Notch Per per
size per D B r d b/T d/t clearance metre tonne
metre t T C N n (mm) (kg) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (m2) (m2) 356 × 406 634 474.7 424.1 47.6 77.0 15.2 290.2 2.75 6.10 551 455.7 418.5 42.0 67.5 15.2 290.2 3.10 6.91 467 436.6 412.4 35.9 58.0 15.2 290.2 3.56 8.08 393 419.1 407.0 30.6 49.2 15.2 290.2 4.14 9.48 340 406.4 403.0 26.5 42.9 15.2 290.2 4.70 11.0 287 393.7 399.0 22.6 36.5 15.2 290.2 5.47 12.8 235 381.0 395.0 18.5 30.2 15.2 290.2 6.54 15.7 26 200 94 2.52 3.98 23 200 84 2.48 4.49 20 200 74 2.42 5.19 17 200 66 2.38 6.05 15 200 60 2.35 6.90 13 200 52 2.31 8.06 11 200 46 2.28 9.70 COLCORE 477 427.0 424.4 48.0 53.2 15.2 290.2 3.99 6.05 26 200 70 2.43 5.09 356 × 368 202 374.7 374.4 16.8 27.0 15.2 290.2 6.93 17.3 10 190 44 2.19 10.8 177 368.3 372.1 14.5 23.8 15.2 290.2 7.82 20.0 9 190 40 2.17 12.3 153 362.0 370.2 12.6 20.7 15.2 290.2 8.94 23.0 8 190 36 2.15 14.1 129 355.6 368.3 10.7 17.5 15.2 290.2 10.5 27.1 7 190 34 2.14 16.6 305 × 305 283 365.3 321.8 26.9 44.1 15.2 246.6 3.65 9.17 240 352.6 317.9 23.0 37.7 15.2 246.6 4.22 10.7 198 339.9 314.1 19.2 31.4 15.2 246.6 5.00 12.8 158 327.2 310.6 15.7 25.0 15.2 246.6 6.21 15.7 137 320.5 308.7 13.8 21.7 15.2 246.6 7.11 17.9 118 314.5 306.8 11.9 18.7 15.2 246.6 8.20 20.7 97 307.8 304.8 9.9 15.4 15.2 246.6 9.90 24.9 15 14 12 10 9 8 7 158 60 1.94 6.85 158 54 1.90 7.93 158 48 1.87 9.45 158 42 1.84 11.6 158 38 1.82 13.3 158 34 1.81 5.3 158 32 1.79 18.4 254 × 254 167 289.1 264.5 19.2 31.7 12.7 200.3 4.17 10.4 132 276.4 261.0 15.6 25.3 12.7 200.3 5.16 12.8 107 266.7 258.3 13.0 20.5 12.7 200.3 6.30 15.4 89 260.4 255.9 10.5 17.3 12.7 200.3 7.40 19.1 13 254.0 254.0 8.6 14.2 12.7 200.3 8.94 23.3 12 10 9 7 6 134 46 1.58 9.44 134 40 1.54 11.7 134 34 1.52 14.2 134 32 1.50 16.9 134 28 1.49 20.3 203 × 203 86 222.3 208.8 13.0 20.5 71 215.9 206.2 10.3 17.3 60 209.6 205.2 9.3 14.2 52 206.2 203.9 8.0 12.5 46 203.2 203.2 7.3 11.0 10.2 160.9 5.09 12.4 10.2 160.9 5.96 15.6 10.2 160.9 7.23 17.3 10.2 160.9 8.16 20.1 10.2 160.9 9.24 22.0 9 108 32 1.24 14.4 7 108 28 1.22 17.2 7 108 26 1.20 20.1 6 108 24 1.19 23.0 6 108 22 1.19 25.8 152 × 152 37 161.8 154.4 8.1 11.5 30 157.5 152.9 6.6 9.4 23 152.4 152.4 6.1 6.8 7.6 123.5 6.71 15.2 6 7.6 123.5 8.13 18.7 5 7.6 123.5 11.2 20.2 5 84 20 0.912 24.6 84 18 0.9 30.0 84 16 0.889 38.7
Table 5.3 Universal columns continued (abstracted from the Steelwork Design Guide to BS 5950: Part 1, published
by the Steel Construction Institute) (b) Properties
Designation Second moment Radius Elastic Plastic Buckling Torsional Warping Torsional Area of area of gyration modulus modulus parameter index constant constant of Serial Mass Axis Axis Axis Axis Axis Axis Axis Axis section
size per x – x y–y x – x y–y x–x y – y x – x y–y u x H J A
metre (mm) (kg) (cm4) (cm4) (cm) (cm) (cm3) (cm3) (cm3) (cm3) (dm6) (cm4) (cm2) 356 × 406 634 275000 98200 18.5 11.0 11 600 4630 14 200 7110 0.843 5.46 38.8 13 700 808 551 227 000 82700 18.0 10.9 9 960 3950 12 100 6060 0.841 6.05 31.1 9 240 702 467 183 000 67 900 17.5 10.7 8 390 3290 10 000 5040 0.839 6.86 24.3 5 820 595 393 147 000 55 400 17.1 10.5 7 000 2720 8 230 4160 7.860.837 19.0 3 550 501 340 122 000 46 800 16.8 10.4 6 030 2320 6 990 3540 0.836 8.85 15.5 2 340 433 287 100 000 38 700 16.5 10.3 5 080 1940 5 820 2950 0.835 10.2 12.3 1 440 366 235 79 100 31 000 16.2 10.2 4 150 1570 4 690 2380 0.834 12.1 9.54 812 300 COLCORE 477 172 000 68 100 16.8 10.6 8 080 3210 356 × 368 202 66 300 23 600 16.0 9.57 3 540 1260 177 57 200 20 500 15.9 9.52 3 100 1100 153 48 500 17 500 15.8 9.46 2 680 944 129 40 200 14 600 15.6 9.39 2 260 790 9 700 4980 3 980 1920 3 460 1670 2 960 1430 2 480 1200 5 100 2340 4 250 1950 3 440 1580 2 680 1230 2 300 1050 1 950 892 1 590 723 2 420 1130 1 870 879 1 490 695 1 230 575 989 462 979 456 802 374 652 303 568 264 497 230 310 140 247 111 0.815 6.91 23.8 5 700 607 0.844 13.3 7.14 560 258 0.844 15.0 6.07 383 226 0.844 17.0 5.09 251 195 0.843 19.9 4.16 153 165 305 × 305 283 240 198 158 137 118 97 78 800 24 500 64 200 20 200 50 800 16 200 38 700 12 500 32 800 10 700 27 600 9 010 22 200 7 270 29 900 9 800 22 600 7 520 17 500 5 900 14 300 4 850 11 400 3 870 9 460 3 120 7 650 2 540 6 090 2 040 5 260 1 770 4 560 1 540 2 220 709 1 740 558 1 260 403 14.8 8.25 4 310 1530 14.5 8.14 3 640 1270 14.2 8.02 2 990 1030 13.9 7.89 2 370 806 13.7 7.82 2 050 691 13.6 7.75 1 760 587 13.4 7.68 1 440 477 0.855 7.65 6.33 2 030 360 0.854 8.73 5.01 1 270 306 0.854 10.2 3.86 734 252 0.852 12.5 2.86 379 201 0.851 14.1 2.38 250 175 0.851 16.2 1.97 160 150 0.850 19.3 1.55 91.1 123 254 × 254 167 132 107 89 73 203 × 203 86 71 60 52 46 152 × 152 37 30 23 11.9 6.79 2 070 741 11.6 6.67 1 630 576 11.3 6.57 1 310 457 11.2 6.52 1 100 379 11.1 6.46 894 305 9.27 5.32 851 299 9.16 5.28 708 246 8.96 5.19 581 199 8.90 5.16 510 174 8.81 5.11 449 151 6.84 3.87 274 6.75 3.82 221 6.51 3.68 166 91.8 73.1 52.9 184 80.9 0.852 8.49 1.62 625 212 0.850 10.3 1.18 322 169 0.848 12.4 0.894 173 137 0.849 14.4 0.716 104 114 0.849 17.3 0.557 57.3 92.9 0.85 10.2 0.317 138 110 0.852 11.9 0.25 81.5 91.1 0.847 14.1 0.195 46.6 75.8 0.848 15.8 0.166 32.0 66.4 0.846 17.7 0.142 22.2 58.8 0.848 13.3 0.04 19.5 47.4 0.848 16.0 0.0306 10.5 38.2 0.837 20.4 0.0214 4.87 29.8
(b) Shear ULS (c) Deflection SLS.
Two other ultimate limit state factors that should be given consideration are:
(d) Web buckling resistance (e) Web bearing resistance.
However, these are not usually critical under normal loading conditions, and in any case may be catered for by the inclusion of suitably designed web stiffeners.
Let us consider how each of these requirements influences the design of beams.
5.10.1 Bending ULS
When a simply supported beam bends, the extreme fibres above the neu-tral axis are placed in compression. If the beam is a steel beam this means that the top flange of the section is in compression and correspondingly the bottom flange is in tension. Owing to the combined effect of the resultant compressive loading and the vertical loading, the top flange could tend to deform sideways and twist about its longitudinal axis as illustrated in Figure 5.3. This is termed lateral torsional buckling, and could lead to premature failure of the beam before it reaches its vertical moment capacity.
Lateral displacement
Original position of beam shown dotted
Lateral displacement shown for half the beam span
Lateral torsional buckling can be avoided by fully restraining the com-pression flange along its entire length (Figure 5.4). Alternatively, trans-verse restraint members can be introduced along the span of the beam (Figure 5.5). These must be at sufficient intervals to prevent lateral tor-sional buckling occurring between the points of restraint. If neither of these measures are adopted then the beam must be considered as laterally unrestrained and its resistance to lateral torsional buckling should be checked. The requirements that must be fulfilled by both lateral and tor-sional restraints are described in BS 5950.
Infill Precast concrete In situ concrete In situ concrete units concrete
Lateral restraint provided by frictional resistance between Shelf angles concrete and steel
Steel beam Steel beam Steel beam
Figure 5.4 Cross-sections through fully laterally restrained beams
Beams connected at right angles to web of main beam Beams connected at right angles to top flange of main beam
Main beam Main beam
Figure 5.5 Cross-sections through beams laterally restrained at intervals along
their length
It can be seen from the foregoing that it is necessary to investigate the bending ULS of steel beams in one of two ways: laterally restrained and laterally unrestrained. These are now discussed in turn.
5.10.2 Bending ULS of laterally restrained beams
It has already been shown in Chapter 1 that, in relation to the theory of bending, the elastic moment of resistance (MR) of a steel beam is given by
MR = ƒZ
A
Beam cross-section Stress diagram
where f is the permissible bending stress value for the steel and Z is the elastic modulus of the section. This assumes that the elastic stress distribution over the depth of the section will be a maximum at the extreme fibres and zero at the neutral axis (NA), as shown in Figure 5.6.
y f D x x D/2 N D/2 f y
Beam cross-section Stress diagram
Figure 5.6 Elastic stress distribution
To ensure the adequacy of a particular steel beam, its internal moment of resistance must be equal to or greater than the applied bending moment:
MR ≥ BM (calculated bending moment) This was the method employed in previous Codes of Practice for steel design based upon permissible stress analysis.
In limit state design, advantage is taken of the ability of many steel sections to carry greater loads up to a limit where the section is stressed to yield throughout its depth, as shown in Figure 5.7. The section in such a case is said to have become fully plastic. The moment capacity of such a beam about its major x–x axis would be given by
Mcx = pySx y D x x y P y N A D/2 Py
Figure 5.7 Plastic stress distribution
where py is the design strength of the steel, given in Table 5.1, and Sx is
the plastic modulus of the section about the major axis, obtained from section tables. In order that plasticity at working load does not occur before the ultimate load is reached, BS 5950 places a limit on the moment capacity of 1.2 pyZ. Thus
Mcx = pySx≤ 1.2 pyZx
The suitability of a particular steel beam would be checked by ensuring that the moment capacity of the section is equal to or greater than the applied ultimate moment Mu:
Mcx
≥
MuThe web and flanges of steel sections are comparatively slender in relation to their depth and breadth. Consequently the compressive force induced in a beam by bending could cause local buckling of the web or flange before the full plastic stress is developed. This must not be confused with the previously mentioned lateral torsional buckling, which is a different mode of failure and will be dealt with in the next section. Nor should it be confused with the web buckling ULS discussed in Section 5.10.6.
Local buckling may be avoided by reducing the stress capacity of the section, and hence its moment capacity, relative to its susceptibility to local buckling failure. In this respect steel sections are classified by BS 5950 in relation to the b/T of the flange and the d/t of the web, where b, d, T and t are as previously indicated in Figures 5.1 and 5.2. There are four classes of section:
Class 1 Plastic Class 2 Compact Class 3 Semi-compact Class 4 Slender.
The limiting width to thickness ratios for classes 1, 2 and 3 are given in BS 5950 Table 7, for both beams and columns. Those for rolled beams are listed here in Table 5.4.
Table 5.4 Beam cross-section classification
Limiting proportions b/T d/t Plastic 8.5 ε 79 ε Class of section Compact 9.5 ε 98 ε Semi-compact 15 ε 120 ε The constant = (275/py)1/2. Hence for grade 43 steel:
When T ≤ 16 mm, ε = (275/275)1/2 = 1 When T > 16 mm, ε = (275/265)1/2
= 1.02 ε
Slender sections are those whose thickness ratios exceed the limits for semi-compact sections. Their design strength p y has to be reduced using a stress reduction factor for slender elements, obtained from Table 8 of BS 5950.
The stress distribution and moment capacity for each class of section is shown in Figure 5.8.
Class 1 Plastic Class 2 Compact Mcx = pySx < 1.2 pyZx Mcx= pySx< 1.2 pyZx
y < py
Class 3 Semi-compact Class 4 Slender M cx = p y Zx Mcx = p'y Zx = kpyZx, where k
is a stress reduction factor from BS 5950, Part 1, Table 8
Figure 5.8 Stress distribution diagrams and moment capacities for section classes
The examples contained in this manual are based upon the use of grade 43 steel sections. All the UB sections formed from grade 43 steel satisfy either the plastic or the compact classification parameters, and hence the stress reduction factor for slender elements does not apply. Furthermore, their plastic modulus S x never exceeds 1.2 times their elastic modulus Zx.
Therefore the moment capacity of grade 43 beams will be given by the expression Mcx = pySx p'y < py py P y P y P y P y py p' = =
By rearranging this expression, the plastic modulus needed for a grade 43 UB section to resist a particular ultimate moment may be determined:
Mu S x required = p
y
Example 5.1
Steel floor beams arranged as shown in Figure 5.9 support a 150 mm thick rein-forced concrete slab which fully restrains the beams laterally. If the floor has to support a specified imposed load of 5 kN/m2 and reinforced concrete weighs
2400 kg/m3, determine the size of grade 43 UBs required.
6 m
Steel floor beams
Slab Slab span span 5 m 5 m Floor plan Ultimate UDL 6 m Isolated steel beam
Figure 5.9 Floor beam arrangement
Before proceeding to the design of the actual beams it is first necessary to calculate the ultimate design load on an individual beam. This basically follows the proce-dure explained in Chapter 1, except that partial safety factors for load γ f need to
be applied since we are using limit state design.
Specified dead load 150 mm slab = 0.15 × 2400/100 = 3.6 kN/m2
Specified imposed load = 5 kN/m2
Specified imposed load UDL = 5 × 6 × 5 = 150 kN
Total ULS design load = γf × specified dead load + γf × specified imposed load = 1.4 × 112 + 1.6 × 150 = 156.8 + 240 = 396.8 kN
WL
Ultimate bending moment Mu =
396.8 × 6 =
8 8
= 297.6 kN m = 297.6 ×106 N mm
The ultimate design strength py for grade 43 steel sections, from Table 5.1, is 275 N/mm2 provided that the flange thickness does not exceed 16 mm. If the flange
thickness was greater than 16 mm, py would reduce to 265 N/mm2. Hence the
plastic modulus is Mu S x required = 297.6 × 10 6 275 = 1 082 182 mm 3 = 1082 cm3 py
It should be appreciated that the plastic modulus property is always tabulated in cm3 units.
By reference to Table 5.2, the lightest UB section with a plastic modulus greater than that required is a 457 × 152 × 60 kg/m UB with an S x of 1280 cm3. It should
be noted that the flange thickness of the selected section is 13.3 mm; this is less than 16 mm, and it was therefore correct to adopt a p y of 275 N/mm2 in the design.
It should also be noted that the self-weight of the section is less than that assumed and therefore no adjustment to the design is necessary; that is,
60
SW = × 6 = 3.6 kN < 4 kN assumed 100
This section would be adopted provided that it could also satisfy the shear and deflection requirements which will be discussed later.
The design approach employed in Example 5.1 only applies to beams which are fully restrained laterally and are subject to low shear loads. When plastic and compact beam sections are subject to high shear loads their moment capacity reduces because of the interaction between shear and bending. Modified expressions are given in BS 5950 for the moment capacity of beams in such circumstances. However, except for heavily loaded short span beams, this is not usually a problem and it will therefore not be given any further consideration here.
5.10.3 Bending ULS of laterally unrestrained beams
Laterally unrestrained beams are susceptible to lateral torsional buckling failure, and must therefore be designed for a lower moment capacity known as the buckling resistance moment M b. It is perhaps worth
reiterat-ing that torsional bucklreiterat-ing is not the same as local bucklreiterat-ing, which also needs to be taken into account by reference to the section classification of plastic, compact, semi-compact or slender.
For rolled universal sections or joists BS 5950 offers two alternative approaches – rigorous or conservative – for the assessment of a member’s lateral torsional buckling resistance. The rigorous approach may be ap-plied to any form of section acting as a beam, whereas the conservative approach applies only to UB, UC and RSJ sections. Let us therefore consider the implications of each of these approaches with respect to the design of rolled universal sections.
Laterally unrestrained beams, rigorous approach
Unlike laterally restrained beams, it is the section’s buckling resistance moment Mb that is usually the criterion rather than its moment capacity Mc. This is given by the following expression:
Mb = pbSx
where pb is the bending strength and Sx is the plastic modulus of the
section about the major axis, obtained from section tables.
The bending strength of laterally unrestrained rolled sections is ob-tained from BS 5950 Table 11, reproduced here as Table 5.5. It depends on the steel design strength py and the equivalent slenderness λLT, which
is derived from the following expression:
LT = nuvλ
where
n slenderness correction factor from BS 5950
u buckling parameter of the section, found from section tables or
con-servatively taken as 0.9
v slenderness factor from BS 5950
minor axis slenderness: = L E/ry
LE effective unrestrained length of the beam
ry radius of gyration of the section about its minor axis, from section tables
The effective length L E should be obtained in accordance with one of the
following conditions:
Condition (a). For beams with lateral restraints at the ends only, the value of L E should be obtained from BS 5950 Table 9, reproduced here as Table
5.6, taking L as the span of the beam. Where the restraint conditions at each end of the beam differ, the mean value of L E should be taken.
Condition (b). For beams with effective lateral restraints at intervals along their length, the value of L E should be taken as 1.0 L for normal loading
conditions or 1.2 L for destabilizing conditions, taking L as the distance between restraints.
Condition (c). For the portion of a beam between one end and the first intermediate restraint, account should be taken of the restraint conditions
λ
Table 5.5 Bending strength p b (N/mm2) for rolled sections (BS 5950 Part 1 1990 Table 11) LT 245 265 275 325 py 340 355 415 430 450 30 245 265 275 325 340 355 408 421 438 35 245 265 273 316 328 341 390 402 418 40 238 254 262 302 313 325 371 382 397 45 227 242 250 287 298 309 350 361 374 50 217 231 238 272 282 292 329 338 350 55 206 219 226 257 266 274 307 315 325 60 195 207 213 241 249 257 285 292 300 65 185 196 201 225 232 239 263 269 276 70 174 184 188 210 216 222 242 247 253 75 164 172 176 195 200 205 223 226 231 80 154 161 165 181 186 190 204 208 212 85 144 151 154 168 172 175 188 190 194 90 135 141 144 156 159 162 173 175 178 95 126 131 134 144 147 150 159 161 163 100 118 123 125 134 137 139 147 148 150 105 111 115 117 125 127 129 136 137 139 110 104 107 109 116 118 120 126 127 128 115 97 101 102 108 110 111 117 118 119 120 91 94 96 101 103 104 108 109 111 125 86 89 90 95 96 97 101 102 103 130 81 83 84 89 90 91 94 95 96 135 76 78 79 83 84 85 88 89 90 140 72 74 75 78 79 80 83 84 84 145 68 70 71 74 75 75 78 79 79 150 64 66 67 70 70 71 73 74 75 155 61 62 63 66 66 67 69 70 70 160 58 59 60 62 63 63 65 66 66 165 55 56 57 59 60 60 62 62 63 170 52 53 54 56 56 57 59 59 59 175 50 51 51 53 54 54 56 56 56 180 47 48 49 51 51 51 53 53 53 185 45 46 46 48 49 49 50 50 51 190 43 44 44 46 46 47 48 48 48 195 41 42 42 44 44 44 46 46 46 200 39 40 40 42 42 42 43 44 44 210 36 37 37 38 39 39 40 40 40 220 33 34 34 35 35 36 36 37 37 230 31 31 31 32 33 33 34 33 34 240 29 29 29 30 30 30 31 31 31 250 27 27 27 28 28 28 29 29 29 λ
Point load applied by column
Column
Main beam (a) Destabilizing detail
Point load applied by column
Column Secondary beams
Main beam (b) Stabilized detail
Figure 5.10 Destabilizing load
Table 5.6 Effective length LE for beams (BS 5950 Part 1 1990 Table 9)
Conditions of restraint at supports Loading conditions Normal Destabilizing Compression flange
laterally restrained Beam fully restrained
against torsion
Both flanges fully restrained against rotation on plan Both flanges partially
restrained against rotation on plan
0.7 L 0.85 L
0.85 L 1.0 L
Both flanges free to rotate 1.0 L 1.2 L on plan
Compression flange Restraint against torsion 1.0 L + 2 D 1.2 L + 2 D
laterally unrestrained provided only by Both flanges free to rotate positive connection of
on plan bottom flange to supports
Restraint against torsion 1.2 L+ 2 D 1.4 L + 2 D provided only by dead
bearing of bottom flange on supports D is the depth of the beam.
L is the span of the beam.
at the support. Therefore the effective length LE should be taken as the
mean of the value given by condition (b) and the value from Table 5.6 elating to the manner of restraint at the support. In both cases, L is taken as the distance between the restrain and the support.
The destabilizing load referred to in the table exists when the member applying the load to the compression flange can move laterally with the beam in question, as illustrated in Figure 5.10a. This may be avoided by the introduction of stabilizing members such as the secondary beams shown in Figure 5.10b.
The slenderness factor v is obtained from BS 5950 Table 14, reproduced here as Table 5.7, using N and /x, where is the slenderness, x is the torsional index of the section from section tables, and N is 0.5 for beams with equal flanges.
To check the adequacy of a particular steel beam section, the buckling moment Mb should be compared with the equivalent uniform moment M :
M ≤ Mb
where M = mMA, m is the equivalent uniform moment factor from BS 5950, and MA is the maximum moment on the member or portion of the member under consideration.
Load
Table 5.7 Slenderness factor v for flanged beams of uniform section (BS 5950 Part 1 1990 Table 14) Compression Compression Tension Tension N 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 / x 0.5 0.79 0.81 0.84 0.88 0.93 1.0 0.78 0.80 0.83 0.87 0.92 1.5 0.77 0.80 0.82 0.86 0.91 2.0 0.76 0.78 0.81 0.85 0.89 2.5 0.75 0.77 0.80 0.83 0.88 1.28 1.57 1.27 1.53 1.24 1.48 1.20 1.42 1.16 1.35 12.67 6.36 4.27 2.20 2.11 1.98 1.84 1.70 3.0 0.74 0.76 0.78 0.82 0.86 3.5 0.72 0.74 0.77 0.80 0.84 4.0 0.71 0.73 0.75 0.78 0.82 4.5 0.69 0.71 0.73 0.76 0.80 5.0 0.68 0.70 0.72 0.75 0.78 1.12 1.29 1.57 1.07 1.22 1.46 1.03 1.16 1.36 0.99 1.11 1.27 0.95 1.05 1.20 3.24 2.62 2.21 1.93 1.71 1.55 1.41 5.5 0.66 0.68 0.70 0.73 0.76 6.0 0.65 0.67 0.69 0.71 0.74 6.5 0.64 0.65 0.67 0.70 0.72 7.0 0.63 0.64 0.66 0.68 0.70 7.5 0.61 0.63 0.65 0.67 0.69 0.92 1.01 1.13 1.31 0.89 0.97 1.07 1.22 0.86 0.93 1.02 1.14 0.83 0.89 0.97 1.08 0.80 0.86 0.93 1.02 8.0 0.60 0.62 0.63 0.65 0.67 8.5 0.59 0.60 0.62 0.64 0.66 9.0 0.58 0.59 0.61 0.63 0.64 9.5 0.57 0.58 0.60 0.61 0.63 10.0 0.56 0.57 0.59 0.60 0.62 0.78 0.83 0.89 0.98 0.76 0.80 0.86 0.93 0.74 0.78 0.83 0.90 0.72 0.76 0.80 0.86 0.70 0.74 0.78 0.83 11.0 0.54 0.55 0.57 0.58 0.60 12.0 0.53 0.54 0.55 0.56 0.58 13.0 0.51 0.52 0.53 0.54 0.56 14.0 0.50 0.51 0.52 0.53 0.54 15.0 0.49 0.49 0.50 0.51 0.52 0.67 0.70 0.73 0.78 0.64 0.66 0.70 0.73 0.61 0.64 0.66 0.69 0.59 0.61 0.63 0.66 0.57 0.59 0.61 0.63 16.0 0.47 0.48 0.49 0.50 0.51 17.0 0.46 0.47 0.48 0.49 0.49 18.0 0.45 0.46 0.47 0.47 0.48 19.0 0.44 0.45 0.46 0.46 0.47 20.0 0.43 0.44 0.45 0.45 0.46 1.00 1.11 0.99 1.10 0.97 1.08 0.96 1.06 0.93 1.03 0.91 1.00 0.89 0.97 0.86 0.94 0.84 0.91 0.82 0.88 0.79 0.85 0.77 0.82 0.75 0.80 0.73 0.78 0.72 0.76 0.70 0.74 0.68 0.72 0.67 0.70 0.65 0.68 0.64 0.67 0.61 0.64 0.59 0.61 0.57 0.59 0.55 0.57 0.53 0.55 0.52 0.53 0.50 0.52 0.49 0.50 0.48 0.49 0.47 0.48 0.55 0.57 0.59 0.61 0.53 0.55 0.57 0.58 0.52 0.53 0.55 0.56 0.50 0.52 0.53 0.55 0.49 0.50 0.51 0.53 Note 1: For beams with equal flanges, N = 0.5; for beams with unequal flanges refer to clause 4.3.7.5 of BS 5950.
Note 2: v should be determined from the general formulae given in clause B.2.5 of BS 5950, on which this table is based: (a) for sections with lipped flanges (e.g. gantry girders composed of channel + universal beam); and (b) for intermediate values to the right of the stepped line in the table.
The factors m and n are interrelated as shown in BS 5950 Table 13, reproduced here as Table 5.8. From this table it can be seen that, when a beam is not loaded between points of lateral restraint, n is 1.0 and m should be obtained from BS 5950 Table 18. The value of m depends upon the ratio of the end moments at the points of restraint. If a beam is loaded between points of lateral restraint, m is 1.0 and n is obtained by reference
Table 5.8 Use of m and n factors for members of uniform section (BS 5950 Part 1 1990 Table 13)
Description Members not subject to destabilizing loads* m n Members subject to destabilizing loads* m n
Members loaded between Sections with equal flanges 1.0 From Tables 15 1.0 1.0 adjacent lateral restraints and 16 of BS 5950
Sections with unequal flanges 1.0 1.0 1.0 1.0 Members not loaded Sections with equal flanges From Table 18 of 1.0 1.0 1.0
between adjacent lateral BS 5950
restraints Sections with unequal flanges 1.0 1.0 1.0 1.0
Cantilevers without intermediate lateral restraints 1.0 1.0 1.0 1.0 *See clause 4.3.4 of BS 5950.
to B3 5950 Tables 15 and 16 (Table 16 is cross-referenced with Table 17). Its value depends upon the ratio of the end moments at the points of restraint and the ratio of the larger moment to the mid-span free moment.
Example 5.2
A simply supported steel beam spans 8 m and supports an ultimate central point load of 170 kN from secondary beams, as shown in Figure 5.11. In addition it carries an ultimate UDL of 9 kN resulting from its self-weight. If the beam is only restrained at the load position and the ends, determine a suitable grade 43 section.
Figure 5.11 Ultimate load diagram
The maximum ultimate moment is given by
WL WL 170 × 8 9 × 8
MA = + + = 340 + 9 = 349 kN m
4 8 4 8
Since the beam is laterally unrestrained it is necessary to select a trial section for checking: try 457 × 152 × 74 kg/m UB (Sx = 1620 cm3). The moment capacity of
this section when the beam is subject to low shear is given by Mcx = pySx, where py is 265 N/mm2 since T is greater than 16 mm. Thus
Mcx = pySx = 265 × 1620 × 103 = 429.3 × 106 N mm = 429.3 kN m > 349 kN m This is adequate. 170 kN UDL 9 kN 4 m 4 m 8 m =
The lateral torsional buckling resistance is checked in the following manner:
M = mMA ≤ Mb = pbSx
The self-weight UDL of 9 kN is relatively insignificant, and it is therefore satisfac-tory to consider the beam to be not loaded between restraints. By reference to Table 5.8, for members that are not subject to destabilizing loads, n is 1.0 and m should be obtained from BS 5950 Table 18.
The values of m in Table 18 depend upon β , which is the ratio of the smaller end moment to the larger end moment for the unrestrained length being considered. In this example the unrestrained length is the distance from a support to the central point load. The bending moment diagram for this length is shown in Figure 5.12.
M = 349 kNm
β M = 0
4 m unrestrained length
Figure 5.12 Equivalent bending moment diagram for the unrestrained length
It can be seen from this diagram that the end moment for a simply supported beam is zero. Hence
β =smaller end moment 0 = 0 larger end moment =349 Therefore the value of m from BS 5950 Table 18 is 0.57.
It should be appreciated that if the central point load was from a column and there were no lateral beams at that point, then a destabilizing load condition would exist. In such a case both m and n, from Table 5.8, would be 1.0.
Equivalent uniform moment M = mMA = 0.57 × 349 = 198.93 kN m
Buckling resistance moment Mb = pbSx
The bending strength pb has to be obtained from Table 5.5 in relation to py and LT. We have py = 265 N/mm
2 and
λ LT = nuv λ
where n = 1.0, u = 0.87 from section tables, and λ = LE/ry. In this instance LE = 1.0 L from Table 5.6, where L is the distance between restraints, and ry = 3.26 cm
= 3.26 × 10 mm from section tables. Thus
λ = 1.0 × 3.26 ×104000= 122.7
Now x = 30 from section tables. Hence /x = 122.7/30 = 4.09. and v = 0.856 by
λ
interpolation from Table 5.7. Hence
LT = nuv = 1.0 × 0.87 × 0.856 × 122.7 = 91.38
Therefore pb = 138.24 N/mm2 by interpolation from Table 5.5. Thus finally Mb = pbSx = 138.24 × 1620 × 103
= 223.95 × 106 N mm = 223.95 kN m > 198.93 kN m
That is, M < Mb. Therefore the lateral torsional buckling resistance of the section
is adequate. In conclusion: Adopt 457 × 152 × 74 kg/m UB.
The Steelwork Design Guide produced by the Steel Construction Institute also contains tables giving both the buckling resistance moment Mb and the moment capacity Mcx for the entire range of rolled sections. A typical example of a number of UB sections is reproduced here as Table 5.9. From the table it can be seen that for the 457 × 152 × 74 kg/m UB section that we have just checked, the relevant moment values are as follows:
Mcx = 429 kN m; and Mb = 223 kN m when n is 1.0 and the effective length is 4.0 m.
By using these tables the amount of calculation is significantly reduced, and they are therefore a particularly useful design aid for checking beams.
Example 5.3
If the beam in Example 5.2 were to be loaded between lateral restraints as shown in Figure 5.13, what size of grade 43 section would be required?
WL WL 170 × 8 40 × 8
MA = = 340 + 40 = 380 kN m4 8+ = +4 8
Figure 5.13 Ultimate load diagram
The maximum ultimate moment at mid-span is given by
It is necessary to select a trial section for checking: try 457 × 191 × 82 kg/m UB (Sx = 1830 cm3). Thus Mcx = pySx = 275 × 1830 × 103 = 503.25 × 106 N mm = 503.25 kN m > 380 kN m This is adequate. 170 kN UDL 40 kN 4 m 8 m 4 m λ λ
Table 5.9 Universal beams subject to bending, steel grade 43: buckling resistance moment Mb(kN m) (abstracted from the Steelwork Design Guide to BS 5950: Part I, published by the Steel Construction Institute)
Designation Slenderness Effective length LE serial size: correction
mass/metre factor and capacity n 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 11.0 457 × 191 × 82 Mcx= 503 Plastic 457 × 191 × 74 Mcx= 456 Plastic 457 × 191 × 67 Mcx= 404 Plastic 457 × 152 × 82 Mcx= 477 Plastic 0 0 0 0 0 0 0 0 457 × 152 × 74 Mcx= 429 Plastic 0 0 0 0 0 0 0 0 457 × 152 × 67 Mcx= 396 Plastic 0 0 0 0 0 0 0 0 457 × 152 × 60 Mcx= 352 Plastic 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 457 × 152 × 52 Mcx= 300 Plastic 406 × 178 × 74 Mcx= 412 Plastic 406 × 178 × 67 Mcx= 371 Plastic 0.4 503 0.6 503 0.8 503 1.0 478 0.4 456 0.6 456 0.8 456 1.0 431 0.4 404 0.6 404 0.8 404 1.0 380 0.4 477 0.6 477 0.8 457 1.0 416 0.4 429 0.6 429 0.8 409 1.0 371 0.4 396 0.6 396 0.8 372 1.0 336 0.4 352 0.6 352 0.8 330 1.0 298 0.4 300 0.6 300 0.8 278 1.0 249 0.4 412 0.6 412 0.8 412 1.0 385 0.4 371 0.6 371 0.8 371 1.0 345 503 503 503 503 503 503 500 478 457 436 480 449 419 389 361 437 396 357 321 289 456 456 456 456 456 456 451 430 410 391 433 404 375 348 321 393 355 319 285 255 404 404 404 404 402 404 397 378 359 341 381 354 328 302 277 345 310 277 246 219 477 477 477 475 462 471 447 424 402 381 422 388 356 326 300 370 327 290 257 231 429 429 429 423 411 421 398 376 355 335 375 343 313 285 260 328 288 252 223 198 396 396 396 384 372 383 361 339 318 299 340 308 278 251 227 294 255 221 193 170 352 352 351 339 328 340 319 299 280 261 301 272 244 219 197 260 224 193 168 147 300 300 295 284 274 286 267 249 231 214 251 225 200 178 158 215 183 156 134 116 412 412 412 412 412 412 405 387 370 354 388 362 337 313 291 351 317 286 257 232 371 371 371 371 370 371 363 346 330 314 347 323 300 277 256 313 282 252 226 202 496 472 451 431 413 417 379 346 317 291 335 289 252 223 199 261 217 184 159 140 446 424 403 384 366 372 337 305 277 253 296 253 219 192 171 230 189 159 137 120 391 370 350 332 314 323 290 260 234 212 254 215 184 160 142 195 159 132 113 98 450 427 407 388 370 362 327 297 272 250 277 238 208 185 167 208 174 149 131 116 399 377 357 339 322 317 284 256 232 212 239 204 177 156 140 178 147 125 109 97 360 338 318 299 283 280 247 220 198 179 207 174 149 130 116 152 124 105 90 79 317 296 276 259 243 244 213 188 167 151 178 148 126 109 96 130 105 87 75 66 263 243 225 208 194 198 170 148 130 116 142 116 97 83 73 102 81 67 57 50 404 386 369 354 339 338 309 283 260 240 270 235 206 183 165 210 175 150 131 116 360 343 327 312 298 299 271 246 225 206 236 203 177 156 139 182 151 128 111 97 395 379 269 249 180 164 126 114 349 333 232 214 154 140 107 96 298 283 194 178 127 114 87 78 326 313 223 208 150 137 105 95 285 273 190 176 126 115 87 79
Mbis obtained using an equivalent slenderness = nuvLe/ry. Values have not been given for values of slenderness greater than 300. The section classification given applies to members subject to bending only.
Check lateral torsional buckling:
M = mMA≤ Mb = pbSx
The magnitude of the UDL in this example is significant, and it will therefore be necessary to consider the beam to be loaded between lateral restraints. By reference to Table 5.8, for members not subject to destabilizing loads, m is 1.0 and n should be obtained from BS 5950 Table 16, which is cross-referenced with Table 17.
First we have
M = mMA = 1.0 × 380 = 380 kN m
We now need to find Mb. The bending strength pb has to be obtained from
Table 5.5 in relation to py and LT. We have py = 275 N/mm2 and LT = nuv
The slenderness correction factor n is obtained from BS 5950 Table 16 in relation to the ratios and ß for the length of beam between lateral restraints. In this instance that length would be from a support to the central point load. The ratios and ß are obtained as follows. First, = M/Mo. The larger end moment M =
380 kN m. Mo is the mid-span moment on a simply supported span equal to the
unrestrained length, that is Mo = WL /8. The UDL on unrestrained length is W =
40/2 = 20 kN, and the unrestrained length L = span/2 = 4 m. Hence
Mo = WL 20× 4 = 10 kNm 8 8=
The equivalent bending moment diagram, for the unrestrained length, corre-sponding to these values is shown in Figure 5.14. Thus
M 380 γ = = = 38M
o 10
Secondly,
ß =smaller end moment 0
larger end moment = = 0380 Therefore, by interpolation from BS 5950 Table 16, n = 0.782.
From section tables, u = 0.877. Mo = 10 kNm
= 380 kNm
ßM = 0
Figure 5.14 Equivalent bending moment diagram for the unrestrained length
2 m 2 m 4 m unrestrained length M λ λ γ γ γ λ
Next = LE/ry, where LE = 1.0 L in this instance from Table 5.6; L is the distance
between restraints; and ry = 4.23 cm from section tables, that is 4.23 × 10 mm. Thus =LrE 1.0 × 4000
y 4.23 × 10
= 94.56
Here x = 30.9 from section tables. Therefore /x = 94.56/30.9 = 3.06, and so
v = 0.91 from Table 5.7.
Finally, therefore,
λ LT = nuv = 0.782 × 0.877 × 0.91 × 94.56 = 59
Using the values of py and LT, pb = 215.6 N/mm2 by interpolation from Table 5.5.
In conclusion,
Mb = pbSx = 215.6 × 1830 × 10
3 = 394.5 × 106 N mm = 394.5 kN m > 380 kN m
Thus M < Mb, and therefore the lateral torsional buckling resistance of the section
is adequate.
Adopt 457 × 191 × 82 kg/m UB.
The Mcx and Mb values that we have calculated may be compared with those tabulated by the Steel Construction Institute for a 457 × 191 × 82 kg/m UB. From Table 5.9, Mcx = 503 kN m, and Mb = 389 kN m when n is 0.8 and the effective length is 4.0.
Laterally unrestrained beams, conservative approach
The suitability of laterally unrestrained UB, UC and RSJ sections may be checked, if desired, using a conservative approach. It should be appreciated that being conservative the design will not be as economic as that given by the rigorous approach; consequently beam sections that are proved to be adequate using the rigorous approach may occasionally prove inadequate using the conservative ap-proach. However, it does have the advantage that members either loaded or unloaded between restraints are checked using one expression.
In the conservative approach the maximum moment Mx occurring between lateral restraints must not exceed the buckling resistance moment Mb:
Mx ≤ Mb
The buckling resistance moment is given by the expression
Mb = pbSx
For the conservative approach, pb is obtained from the appropriate part of Table 19 a–d of BS 5950 in relation to and x, the choice depending on the design strength py of the steel.
Loads occurring between restraints may be taken into account by multiplying the effective length by a slenderness correction factor n obtained either from BS 5950 Table 13 (reproduced earlier as Table 5.8) or alternatively from BS 5950 Table 20, except for destabilizing loads when it should be taken as 1.0. It is important to understand that the reactions shown on the diagrams in Table 20 are the lateral restraints and not just the beam supports. Therefore for a simply supported beam with a central point load providing lateral restraint, the relevant Table 20 diagram would be as shown in Figure 5.15. The corresponding value of
n would then be 0.77. = λ λ λ λ λ λ
Central point load Support Unrestrained length
(a) Portion of beam between (b) Corresponding bending lateral restraints moment diagram
Figure 5.15 Conservative approach slenderness correction factor diagrams for a
simply supported beam restrained at mid-span
Thus the minor axis slenderness ratio is given by
nLE =
ry
where n is the slenderness correction factor either from BS 5950 Table 13 or Table 20, LE is the effective unrestrained length of the beam, and ry is the radius of
gyration of the section about its minor axis, found from section tables. The torsional index x of the section is taken from section tables.
For those who are familiar with BS 449, this approach is similar to the use of Table 3 in that standard, which was related to the l/r and D/T ratios of the section.
Example 5.4
Check the beam section selected in Example 5.3, using the conservative approach. The maximum ultimate moment Mx = 380 kN n at midspan. Check 457 ×
191 × 82 kg/m UB (Sx = 1830 cm3). T = 16mm; hence py = 275 N/mm2. Thus Mcx = pySx = 275 × 1830 ×103 = 503.25 × 106 N mm = 503.25 kN m > 380 kN m
This is satisfactory:
Check lateral torsional buckling, that is show
Mx ≤ Mb = pbSx
For the conservative approach, pb is obtained from BS 5950 Table 19b when py is 275 N/mm2, using and x. The slenderness correction factor n obtained from
BS 5950 Table 20 is 0.77. Then
nLE 0.77 × 4000 = =
ry 4.23 ×10 = 72.8
Now x = 30.9. Thus pb = 210 N/mm2 by interpolation from BS 5950 Table 19b. So
Mb = pbSx = 210 × 1830 × 10 3
= 384.3 × 106 N mm = 384.3 kN m > 380 kN m Therefore Mx < Mb, and so the lateral torsional buckling resistance of the section is adequate.
M
λ
λ λ
